The electrical properties of monolithic microwave integrated circuits that are connected to transmission lines are described in terms of their scattering matrix using Maxwell's equations. Using a finite-volume method the corresponding three-dimensional boundary value problem of Maxwell's equations in the frequency domain can be solved by means of a two-step procedure. An eigenvalue problem [1] for non-symmetric matrices yields the wave modes. The eigenfunctions determine the boundary values at the ports of the transmission lines for the calculation of the fields in the three-dimensional structure. The electromagnetic fields and the scattering matrix elements are achieved by the solution of large-scale systems of linear equations with indefinite complex symmetric coefficient matrices. In many situations, these matrix problems need to be solved repeatedly for different right-hand sides, but with the same coefficient matrix. The block quasi-minimal residual algorithm [2] is a block Krylov subspace iterative method that incorporates deflation to delete linearly and most linearly dependent vectors in the underlying block Krylov sequences.

[1] G. Hebermehl, R. Schlundt, H. Zscheile, W. Heinrich. Improved numerical methods for the simulation of microwave circuits. Surveys on Mathematics for Industry, Vol. 9, No. 2, pp. 117-129, 1999.
[2] R.W. Freund, M. Malhotra. A block-QMR algorithm for non-Hermitian linear systems with multiple right-hand sides. Linear Algebra and Its Applications, Vol. 254, pp. 119-157, 1997.